For most of my life, one single (family of) Diophantine equation(s) dominated the list of the world's most celebrated unsolved mathematical problems. Perhaps the world we live in now has grown too sophisticated for such heavy focus on such a narrow question (visions of Fermat's MO postings getting closed as "Too localized"), but if not:

What specific unsolved Diophantine equations would today's number theorists most like to crack? (And why - historical provenance, application to another part of mathematics, "test" question for a major arithmetical theory, etc.)

"Specific" means, for example, "solve BSD, etc. to find an algorithm to decide the solvability of all elliptic curves" doesn't count, nor is this is the place to talk about anything like the $abc$ conjecture.

Answers should not depend on any sort of coding, Matiyasevich-style.

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    $\begingroup$ The following two questions (which seem to be near-duplicates of one another) are also closely related to your question: mathoverflow.net/questions/21012/… mathoverflow.net/questions/42406/… $\endgroup$ – Emerton Feb 9 '11 at 8:35
  • $\begingroup$ Excellent point, Emerton. One should certainly be closed as duplicate, or merged into the other (which I believe is possible). It is unfortunate that the one that was posted first received less attention and only one answer; that user should get the credit for it. $\endgroup$ – Zev Chonoles Feb 9 '11 at 9:45
  • $\begingroup$ Thanks Emerton and Zev, I find the answers to those question helpful. My question does have a different thrust. I'm asking specifically for open problems that have actually acquired some currency or urgency among those who do research in the area, rather than for general reflection on what might make an equation interesting to study. $\endgroup$ – David Feldman Feb 9 '11 at 13:37

You may be interested in Some open problems about diophantine equations; it says, "We have collected some open problems which were posed by participants of an instructional conference (May 7-11, 2007) and a subsequent more advanced workshop (May 14-16, 2007) on solvability of Diophantine equations, both held at the Lorentz Center of Leiden University, The Netherlands. Among the 22 problems posed:

Find all integer solutions to the equation $x^2-x=y^5-y$.

Do the same for the equation ${x\choose2}={y\choose5}$.

Determine all arithmetic progressions of the form $a^2,b^2,c^2,d^5$ where $a,b,c,d$ are integers with $\gcd(a,b)=1$.

Are there infinitely many positive integer solutions to ${x^3-1\over y^3-1}=z^2$?

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    $\begingroup$ What is the motivation for these problems? Is it something deeper than "we don't know how to answer them"? $\endgroup$ – rghthndsd Jun 11 '14 at 2:24
  • $\begingroup$ The problems are presented without motivation. I think you would have to ask the people who posed them what motivation there might be. $\endgroup$ – Gerry Myerson Jun 11 '14 at 3:32
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    $\begingroup$ The equations $x^2-x=y^5-y$ and ${x\choose2}={y\choose5}$ are solved here: arxiv.org/abs/0801.4459 $\endgroup$ – Siksek Jun 11 '14 at 8:08
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    $\begingroup$ The arithmetic progressions of the form $a^2$, $b^2$, $c^2$, $d^5$ where $a$, $b$, $c$, $d$ are integers with $\gcd(a,b)=1$ are determined here: arxiv.org/abs/0912.2670 $\endgroup$ – Siksek Jun 11 '14 at 8:12
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    $\begingroup$ The motivation for theses is problems is that they were beyond what known methods for Diophantine equations were capable of at the time. This is still true of the last problem above: $(x^3-1)/(y^3-1)=z^2$. This problem asks about integral points on a surface. Our understanding of the arithmetic of surfaces is still embryonic. $\endgroup$ – Siksek Jun 11 '14 at 8:16

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